We consider the problem of pricing contingent claims using
distortion operators. This approach was first developed in (Wang, 2000)
where the original distortion function was defined in terms of the
normal distribution. Here, we introduce a new distortion based on the
Normal Inverse Gaussian (NIG) distribution. The NIG is a generalization
of the normal distribution that allows for heavier skewed tails. The
resulting operator asymmetrically distorts the underlying distribution.
Moreover, we show how we can recuperate non-Gaussian Black-Scholes
formulas using distortion operators and we provide illustrations of
their performance. We conclude with a brief discussion on risk
management applications.

INTRODUCTION

In Wang (2000), the author proposes a form of insurance risk
pricing based on a normal-based distortion risk measure. Distortion risk
measures are quantile-based measures that have been developed in the
actuarial literature and that are now part of the risk measurement tools
inventory available for practitioners in finance and insurance (see Dowd
and Blake, 2006, for an account on these and other risk measures). It
turns out that this distortion-based pricing principle is consistent
with the financial theory of Gaussian option pricing. In Hamada and
Sherris (2003), it is shown that the celebrated Black-Scholes formula
can be recuperated through the distortion operator of Wang (2000) under
the assumption of a normal model for asset prices. Moreover, the authors
carry out a numerical analysis of the normal-based distortion operator
of Wang (2000) in order to assess its performance under a non-normal
model for asset prices. They numerically illustrate the limitations of
Wang's approach under non-Gaussian assumptions. Another downside of
the normal distortion of Wang (2000) is its underlying symmetry that
poses some constrains in applications. In Wang (2004), we find an
application in catastrophe (CAT) bonds pricing where a Student-t
distribution-based distortion is introduced. Unlike the normal operator
of Wang (2000), this Student-t distortion allows for skewness that
translates into large losses as well as large gains being inflated under
the distorted probability. In this article, we address the concerns in
Hamada and Sherris (2003) while using a distortion that brings skewness
into the picture. Indeed, this new family of distortion, which is based
on an asymmetric distribution, allows for similar applications as those
discussed in Wang (2004).

It is a well-known fact that the returns of most financial assets
have semi-heavy tails and the actual kurtosis is higher than that of a
normal distribution. Indeed, a large body of literature has documented
common features as skewness and excess kurtosis of asset returns
(Bollerslev, 1987; Richardson and Smith, 1993; Ghose and Kroner, 1995;
McCulloch, 1997; Theodossiou, 1998; Rockinger and Jondeau, 2002; Jondeau
and Rockinger, 2003; Theodossiou and Trigeorgis, 2003; Bali and
Theodossiou, 2007; Bali and Weinbaum, 2007; among others). In particular
Bali (2003) provides this evidence using the extreme value
distributions. Moreover, several studies have suggested different
distributions to capture the fat tails of asset returns. These include
the Student-t (Bollerslev, 1987; Hsieh, 1989), Generalized t
distribution (McDonald and Neweys, 1988), skewed Generalized t (McDonald
and Newey, 1988; Theodossiou, 1998; Bauwens and Laurent, 2002),
non-central-t distribution by Harvey and Siddique (1999), SU-normal
distribution (Pilsun and Nam, 2008), exponential generalized beta of the
second kind (Wang et al., 2001), and Pearson Type IV (Nagahara, 1999).

Clearly, these stylized features of asset returns have to be taken
into account in risk measurements. For instance, in Bali and Theodossiou
(2008) we find a recent study where non-Gaussian distributions are used
for VaR and TVaR estimation. Similarly, the pricing principle approach
of Wang must be somehow modified in order for it to capture the
non-Gaussian feature of market prices. In recent years, several
non-Gaussian distributions have been proposed in order to better model
asset prices. Recently, the Generalized Hyperbolic family of
distributions has been successfully used in finance (Eberlein and
Keller, 1995; Eberlein, 2001; Prause, 1999). This family has many
interesting properties that better capture the semi-heavy tail feature,
skewness, and kurtosis of financial returns. It is a very large family
that contains the normal distribution as a limiting case. Among these
distributions, we find the subclass of Normal Inverse Gaussian (NIG)
distributions. This law is a natural generalization of the normal
distribution and, with four parameters, is more flexible in terms of
skewness and kurtosis. Just as the normal distribution is the
cornerstone for the geometric Brownian motion, we can construct an
exponential NIG Levy motion that is a natural generalization of the
former. Based on models like the NIG process, a non-Gaussian financial
theory is now readily available (see Schoutens, 2003, for an
introduction). Under non-Gaussian assumptions for the asset returns,
markets are incomplete and there are many equivalent martingale
measures. This implies in turn that the arbitrage-free price of
contingent claims is not unique. Among this family of equivalent
martingale measures, one can find subclasses for which explicit formulas
can be worked out. One such subclass is the so-called mean-correcting
equivalent martingale measure. Within this class of equivalent measures,
we have that a NIG-Levy processes remains within the same family of
processes with only a change in the mean parameter. Moreover, a
Black-Scholes type formula can be worked out in this case. This is an
interesting analog to the situation found in the Brownian model for
which the celebrated Black-Scholes formula for contingent claims was
first developed.

In Hamada and Sherris (2003), they set out to illustrate the
limitations of the operator of Wang under non-Gaussian assumptions. In
this note, we propose a generalized version of this operator based on a
NIG distribution. We show that this operator is compatible with standard
non-Gaussian financial theory and in particular, we recuperate the
option formula under the mean-correcting subclass of equivalent
martingale measures. We also carry out a simulation analysis in order to
illustrate how this new distortion operator improves upon the
limitations of Wang's original distortion, in particular, those
discussed in Wang (2004). Following Hamada and Sherris, we consider four
asset price models: a geometric Brownian motion, an exponential NIG-Levy
model, a jump diffusion model, and a constant elasticity of variance
(CEV) model. We compare the performance of the normal-based operator
(Wang, 2000), the asymmetric Student-t operator of Wang and our new
NIG-based operator for all four models. We confirm again that
Wang's normal operator does not perform very well under
non-Gaussian assumptions. But we also show that the NIG-based operator
is more robust under different models for the underlying even under
different non-Gaussian assumptions.

The article is organized as follows. In "The Normal Inverse
Gaussian Distribution and Non-Gaussian Black-Scholes Contingent
Pricing" section, we present a brief summary of results about the
NIG family of distributions and the corresponding non-Gaussian financial
theory. We also introduce in this section several results that will be
needed throughout out this article. In the "A New Class of
Distortion Operators" section, we introduce a NIG-based distortion
operator and discuss some of their properties and features. In the
"Contingent Claims Pricing" section, we show how this new
operator is consistent with standard non-Gaussian financial theory by
recuperating the Black-Scholes type formula. In the "Simulation and
Further Analysis" section, we present a simulation analysis of this
operator for several examples. Finally, in the "Conclusions and
Further Discussion" section, we conclude with a brief discussion
some potential applications in risk management while hinting at
interesting directions for future work.

The NIG distribution is a member of the wider class of generalized
hyperbolic distributions. This larger family was introduced in
Barndorff-Nielsen and Halgreen (1977). It contains either directly or as
a limiting case the inverse Gaussian, normal, Student-t, Cauchy,
exponential, and gamma distributions. In the last decade, this family
has been widely applied in finance (see Eberlein and Keller, 1995;
Eberlein, 2001; Prause, 1999). It is a well-known fact that the returns
of most financial assets have semi-heavy tails and the actual kurtosis
is higher than that of a normal distribution. Recent studies propose the
generalized hyperbolic family of distributions as a more adequate model
for asset returns. They show that the medium-tailed generalized
hyperbolic distribution fits well to stock returns and use it in a
general option pricing model. In financial applications it is often
preferred to [alpha]-stable distributions since their density is known
and all of the moments exist. Moreover, it has been shown
(Barndorff-Nielsen and Halgreen, 1977) that this family belongs to the
infinitely divisible class of distributions that allows for the
construction of a non-Gaussian Black-Scholes option pricing theory. In
this note we focus on the NIG subclass of this large family, a thorough
account of more general non-Gaussian distributions in finance can be
found in Schoutens (2003).

In this section, we present some well-known facts and results about
the NIG subclass and its applications in finance. The NIG is one of the
only two subclasses being closed under convolutions (the other one being
the variance-gamma distribution). Its density function is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [K.sub.[lambda]] is the modified Bessel function of the third
kind with index [lambda] given by

An interesting feature of the NIG density is that, unlike the
normal density, it is not symmetric and that its asymmetry is determined
by the parameter [beta] as it can be inferred from the density itself
(1) where [beta] appears in the exponential term. A positive [beta]
leads to right skewness whereas a negative [beta] has the opposite
effect. Moreover, it is a straightforward exercise to verify that the
NIG distribution has a symmetry property with respect to fl as stated in
the following remark.

Remark 2: For a given parameter [beta] such that [[alpha].sup.2]
> [[beta].sup.2], we have that

This form of the Laplace transform yields an expression for the
expectation of an exponential transformation of a NIG random variable.
This is given in the following remark.

Remark 3: If X ~ nig([alpha], [beta], [delta], [mu]) we have that Y
= [e.sup.X] is a LogNIG random variable with mean given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the form of the Laplace transform (2), we can see that the NIG
distribution is closed under convolutions. If [X.sub.1] and [X.sub.2]
are two independent random variables with NIG densities nig(x; [alpha],
[beta], [[delta].sub.1], [[mu].sub.1]) and nig(x; [alpha], [beta],
[[delta].sub.2], [[mu].sub.2], respectively, then [X.sub.1] + [X.sub.2]
has density nig(x; [alpha], [beta], [[delta].sub.1], [[mu].sub.1] +
[[mu].sub.2]). Moreover, the NIG is closed under affine transformations
as stated in the following remark.

Remark 4: If X ~ nig([alpha], [beta], [delta], [mu]) we have that Y
= aX + b, for a > 0 and b [member of] R, is such that

Y ~ nig([alpha]/a, [beta]/a, a[delta], a[mu] + b).

The NIG distribution was originally constructed in
Barndorff-Nielsen (1977) as a normal variance-mean mixture where the
mixing distribution is an inverse Gaussian. This is, if X is a NIG
distributed random variable then, the conditional distribution given W =
w is N([mu] + [beta]w, w) where W is inverse Gaussian distributed
IG([delta], [square root of [[alpha].sup.2] - [[beta].sup.2])] (see
Jorgensen, 1982, for a reference on inverse Gaussian distributions).
This gives a simple way of simulating NIG random variables. Moreover,
this property allows us to numerically evaluate the NIG distribution
function through the following result that is now standard in the
literature (see, e.g., Schoutens, 2003).

Proposition 1: The distribution function NIG is given by the
expression

where [PHI] is the standard normal distribution and the inverse
Gaussian density ig is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Barndorff-Nielsen and Halgreen (1977) showed that the generalized
hyperbolic family is infinitely divisible. Because of the infinite
divisibility of NIG distributions, we can construct a NIG Levy process,
that is, a process with independent and stationary NIG-distributed
increments. Recall that this class of processes is in one-to-one
correspondence with the class of infinitely divisible distributions.
Every infinitely divisible distribution generates a Levy process and the
increments of every Levy process are infinitely divisible distributed.
We refer to Bertoin (1996) or Sato (1999) for comprehensive discussions
on Levy processes and to Barndorff-Nielsen, Mikosh, and Resnick (2001)
for recent applications.

The NIG process is an extension of the Brownian motion that allows
for finite dimensional distributions with semi-heavy tails. In a way, it
can be seen as a purely discontinuous version of the latter. Within the
wide spectrum of Levy processes, it lies somewhere between the Brownian
motion and the [alpha]-stable process. In finance, Brownian motion is at
the heart of the Black-Scholes option pricing theory. In recent years,
this arbitrage-free analysis has been extended to include more general
Levy processes as the basic building block. In this section, we briefly
mention some elements of this so-called non-Gaussian Black-Scholes
option pricing. For a more extensive introduction we refer the reader to
Schoutens (2003).

The NIG process is a pure jump process plus a drift term. The drift
term is nothing but the expected value of X(1). In Figure 1 we can see
different paths of NIG processes.

The NIG Levy process exhibits a diffusion-like feature along with a
jump-driven structure. Despite the apparent continuity, its paths are
composed by an infinite number of small jumps. We refer to Prause (1999)
for a comprehensive discussion on the NIG Levy process.

In all, the NIG process has features that make it a natural model
for asset returns. A general exponential model for asset prices in terms
of a NIG process has been proposed and analyzed in Barndorff-Nielsen
(1998). The so-called exponential NIG Levy model for asset prices is of
the form

[S.sub.t] = [S.sub.0e][Z.sub.t], t > 0, (4)

where [Z.sub.t] is a ([F.sub.t], P)-NIG Levy process. A
Black-Scholes analysis can still be carried out for derivatives defined
on a more sophisticated market composed by a risky asset following (4)
and a riskless asset of the form [B.sub.t] = [e.sup.rt]. The fundamental
theorem of asset pricing still applies and the arbitrage-free price of a
derivative is still given in terms of an expectation under a
risk-neutral (equivalent) probability measure (Delbaen and
Schachermayer, 1994). This is, an European-type contingent payment
f([S.sub.T]) has an arbitrage-free price [C.sub.t] given by

where Q is an probability measure, equivalent to P, under which the
discounted asset [e.sup.-rt][S.sub.t] is a ([F.sub.t], Q)-martingale. It
turns out such markets are incomplete and therefore, unlike the
geometric Brownian motion case, there exist an infinite number of such
equivalent martingale measures (Eberlein and Jacod, 1997). Despite the
lack of uniqueness of the equivalent martingale measure, if we restrict
ourselves to a certain family of equivalent measures, closed-form
formulas can be obtained. A common choice is the mean-correcting
martingale measure (see Schoutens, 2003, for a thorough description).
Under this change of measure, the original drift parameter [mu] becomes
[mu] + [[theta].sup.*], therefore, the name mean correcting. Within this
family of equivalent measures, it can be shown that a Black-Scholes-like
formula can be obtained for a European-type contingent claim. This fact
is stated in the following result.

[FIGURE 1 OMITTED]

Theorem 1: Let St be the exponential NIG-Levy price process defined
in (4) with parameters [[alpha], [beta], [delta], [mu]]. One possible
arbitrage-free price of a European-type contingent pay-off f([S.sub.T])
at time t is given by

where [Q.sub.[theta]*] is an equivalent martingale measure under
which St is an exponential NIG-Levy process with parameters [[alpha],
[beta], [delta], [mu + [[theta].sup.*]] and the value of [[theta].sub.*]
is given by

Proof: This is a standard result in the literature. We refer to
Schoutens (2003) for more details. Q.E.D.

We remark that in this setting the parameter [mu] plays the role of
the drift in the Black-Scholes setting. We notice that for this
equivalent measure to exist we need the condition [([beta] + 1).sup.2]
< [[alpha].sup.2].

where [[theta].sup.*] is given in (7). We remark that, just like in
the classical Black-Schole setting, the call option pricing Formula (8)
does not depend in [mu]. Indeed, a closer look at Equation (8) shows
that the parameter [mu] cancels out with the one appearing inside the
parameter [[theta].sup.*].

Formula (8) has the same structure as the Black-Scholes formula and
it is just as simple to compute. It only requires the evaluation of the
distribution function of a NIG. This can be easily implemented through
Equation (3).

For a comprehensive treatment of equivalent martingale measures and
non-Gaussian option pricing we refer the reader to Prause (1999) and
Schoutens (2003). In the following section, we discuss how we can define
a distortion operator based on a NIG distribution.

A NEW CLASS OF DISTORTION OPERATORS

In Wang (2000), the author discusses an approach to price financial
and insurance risks alike. This approach stems out of the dual theory of
risk developed in the economic literature (Yaari, 1987). Under this
approach, the risk premium (or price) of an insurance (or financial)
position is given in terms of a distortion function g that somehow
encapsulates the perceived risk aversion. Let X be a random variable
representing a financial (insurance) risk and let [F.sub.X] and
[S.sub.X] be its distribution and survival function respectively. The
premium (price) associated with this position is

[PI](X) = [integral] g ([S.sub.X](x)) dx, (9)

where g is an increasing differentiable function with 0 [less than
or equal to] g(x) [less than or equal to] 1 for all x. Notice that all
integrals in this section have to be understood as integrals over the
entire domain of the random variable X. Moreover, this function is such
that g(0) = 0 and g(1) = 1. This function is a so-called distortion
function. A close look at Equation (9) shows that the premium function
1-I can be seen as a corrected mean under a new density measure given by
[eta](dx) = g'([S.sub.X](x))d[F.sub.X](x), that is,

where [THETA] is the standard normal cumulative distribution
function and [[PSI].sub.k] is a Student-t cumulative distribution with k
degrees of freedom.

This distortion allows for asymmetric distortion of the tail
probabilities. We refer to the original article Wang (2004) for a
comprehensive discussion on these features.

Later in this section, we introduce a new class of distortion
functions that improves upon the original distortion (11) in a similar
fashion as the more recent class of operators in (12). In addition, a
numerical comparison of all three distortions is carried out. For now,
we focus on the consistency of financial pricing using distortion
operators and the classical Black-Scholes framework.

In Hamada and Sherris (2003), they show that the distortion in (11)
is consistent with the Black-Scholes formula. Let us consider the
following pricing kernel associated to the distortion (11),

H[X = h(Z);[alpha]] = [integral] [g.sub.[alpha]]([S.sub.X](x)) dx,

where h is a continuous, positive and increasing function. It is a
straightforward exercise to show that, for a normal random variable Z

H[X = h(Z);[alpha]] = E[h(Z + [alpha])].

In the standard Black-Scholes model, the risk position at time
[X.sub.t] is modeled by a geometric Brownian motion, that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where W is a standard Brownian motion.

If we consider the payoff of a European call option (with maturity
T and strike price K), we can write

C([X.sub.T], K) = [([X.sub.T] - K).sub.+],

where XT is a log-normal random variable. If we now apply the
pricing kernel to this payoff with [[alpha].sup.*] - ([mu] - r)[square
root of (T)]/[sigma], we can show that

and r the market risk-free rate. That is, we can recuperate the
Black-Scholes formula through the distortion operator (11).

Along the same lines, we introduce a generalized version of the
distortion (11) that is based on a NIG distribution rather than on a
normal distribution. Moreover, we show that the mean-correcting
non-Gaussian Black-Scholes formula can be recuperated from this new
NIG-based distortion operator.

We notice that unlike the distortions in (11) and (12), this
generalized distortion has four parameters that can be calibrated from
data just like the [alpha] parameter in Wang's distortion (11) and
the [alpha] and k parameters in (12). In fact, for computational
purposes the calibration for these parameters can be reduced to
calibrating only three using the following reparametrization [xi] =
[square root of ([alpha][delta])] and [zeta] = - [beta][square root of
([delta]/[alpha])]. In all, these extra parameters give more flexibility
to this family of distortions as we will illustrate in the
"Simulation and Further Analysis" section. Indeed, the
resulting risk-adjusted probability can be thought of as containing a
risk premium for higher moments through these parameters. In particular,
the skewness of any data set will be captured through the parameter
[beta]. This is in fact an interesting feature of this new distortion.
Since it is based on a skewed distribution, the underlying probabilities
are distorted asymmetrically at the tails through the parameter [beta].
In Wang's distortion, both tails are distorted in the same way
because of the symmetry of the normal distribution. In this respect, the
NIG distortion (13) is somewhat similar to the asymmetric distortion
(12).

We start our discussion with the following proposition that shows
the effect of the NIG distortion on a NIG random variable.

Proposition 2: Consider the NIG distortion
[g.sub.[alpha],[beta],[delta],0] defined in (13). Let Z be a random
variable with distribution given by NIG([alpha], [beta], [delta], [mu])
and let X = h(Z) be a transformation through a continuous, positive and
increasing function h. Then,

Because of the symmetry property of the parameter [beta] in Remark
2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[THETA].sup.NIG] denotes the cumulative distribution
function NIG([square root of [alpha][delta])], [beta][square root of
([delta]/[alpha])], [square root of ([alpha][delta])], 0). If we now
apply the distortion [g.sub.[alpha],[beta],[delta],[theta]] on [S.sub.X]
we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we use the symmetry property of the [beta] parameter again we
have

where Y ~ NIG([square root of ([alpha][delta])], [beta] [square
root of ([delta]/[alpha]], [square root of ([alpha][delta])], 0). Using
Remark 4, we can see that [square root of ([delta]/[alpha])] Y + mu ~
NIG([alpha], [beta], [delta], [mu]), so that we can write

This result shows that under a
[g.sub.[alpha],[beta],[delta],[theta]] distortion, a NIG random variable
is translated by a factor [theta] [square root of ([delta]/[alpha]].
This generalizes the equivalent result found in Hamada and Sherris
(2003).

We now have to study how this distortion affects an exponential
Levy model for asset prices and in particular if there is a value of
[theta] such that discounted asset prices behave like risk-neutral asset
prices. Let us consider the following exponential NIG asset price model

[S.sub.t] = [S.sub.0][e.sup.Zt], t > 0, (15)

where [Z.sub.t] is a ([F.sub.t], P)-NIG Levy process with
parameters [[alpha], [beta],[delta], [mu]]. Then, the ([F.sub.T],
P)-random variable [S.sub.T] is the price of the security at time T and
it can be written as [S.sub.T] = h([Z.sub.T]) for a function h(u) =
[S.sub.0][e.sup.u] and a random variable [Z.sub.T] with distribution
NIG([alpha], [beta], [delta]T, [mu]T).

If we apply Proposition 2 we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This comes from Remark 4, which implies that [Z.sub.T] - [theta]
[square root of ([delta]T/[alpha])] ~ NIG([alpha], [beta], [delta]T,
[mu]T -- [theta] ([delta]T/[alpha])] and from the form of the
expectation of a LogNIG random variable give in Remark 3.

In other words, under the NIG distortion with a value of [theta]
given in (16), the price [S.sub.T] evolves like a risk-neutral asset.

CONTINGENT CLAIMS PRICING

In this section, we show how the NIG distortion operator defined in
the "A New Class of Distortion Operators" section is
consistent with the non-Gaussian option pricing theory. In particular,
we recuperate the non-Gaussian Black-Scholes option pricing Formula (8).

Let us consider the NIG asset model in (15) and a standard European
call option payoff at maturity T given by f([S.sub.T], K) = [([S.sub.T]
- K).sub.+]. This is clearly a function of the ([F.sub.T], P)-random
variable ST that is the price of the security at time T. It can be
written as [S.sub.T] = h([Z.sub.T]) for a function h(u) =
[([S.sub.0][e.sup.u] - K).sub.+] and a random variable [Z.sub.T] with
distribution NIG([alpha], [beta], [delta]T, [mu]T).

If we apply Proposition 2 we have that the price of this standard
European call payoff is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

This comes from Remark 4 and from the fact that [Z.sub.T] ~
NIG([alpha], [beta], [delta]T, [mu]T). The values of z for which
[S.sub.0][e.sup.z] - K > 0 is the interval (In K/[S.sub.0],
[infinity]), and the integral in (17) becomes,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

If we set the parameter [theta] to be the one that makes [S.sub.T]
evolve like a risk-neutral asset, that is,

The first integral in (19) can be reduced to a much simpler form by
directly using the expression in (1) for the NIG density. This yields,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that the price of a standard European pay-off
evaluated with the pricing kernel associated to the NIG distortion (13)
with a parameter 0* is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

Equation (20) is the Black-Scholes-type formula in (8) at t = 0.
This shows that the NIG-distorted pricing kernel in (13), with parameter
[[theta].sup.*], reduces to the Black-Scholes-type formula under the
mean-correcting martingale measure.

SIMULATION AND FURTHER ANALYSIS

In this section, we carry out an empirical analysis of the
performance of a pricing kernel based on (13) under different types of
data sets. Our aim is two-fold. First, we want to corroborate the
findings in Hamada and Sherris (2003) according to which the Wang
distortion operator (11) performs poorly when it is used outside a
Gaussian setting. This is a major drawback since it is a well-documented
fact that market log-returns are anything but Gaussian. Our second aim
is to show how the NIG distortion operator in (13) performs better in a
wide range of non-Gausian situations. This is an interesting feature of
the NIG operator since one would expect to use such a pricing kernel on
actual market asset prices and not on Gaussian examples.

In order to provide a controlled setting for our analysis,
simulated data are used to test all three distortions discussed in this
article, namely, the normal distortion (11), the Student-t distortion
(12) and the NIG distortion (13). These pricing kernels are tested on
four different simulated data sets. Asset prices are simulated according
to four well-known models for which theoretical option price formulas
are available: a geometric Brownian model, a log-normal model with
jumps, a constant elasticity of variance (CEV) model and a NIG-Levy
model. Both pricing kernels were tested on these data sets in order to
compare how these two distortions perform when evaluating options on
underlying assets that do not have Gaussian log-returns.

Let X = C([S.sub.T], K) be the European call pay-off with strike K.
These options are evaluated through Equation (9) using all three
distortions in (11), (12), and in (13) under the four different models.

WANG DISTORTIONS

First, for all four models, the performance of both Wang's
distortion operators (11) and (12) are evaluated. Using the simulated
data, the empirical survival function [[??].sub.X] is computed. The
parameter [alpha] is then calibrated to verify the risk-neutral
condition such that

H[[S.sub.T]; - [alpha]] = [S.sub.0][e.sup.rT].

The latter is then used in order to compute the right-hand side in
(9) with both distortions (11) and (12), that is,

The performance of the NIG distortion is also tested on the four
models. Using the simulated data, the empirical survival function
[[??].sub.X] is computed. The latter is then used in order to compute
the right-hand side in (9) with the NIG distortion in (13), that is,

Here, parameters [alpha], [beta], and [delta] have to be first
estimated from log-returns of simulated data. The estimation is done by
fitting the NIG distribution to the simulated log-returns using maximum
likelihood estimation. This estimations is implemented using the nigFit
tool in R. Alternatively, we can refer to an algorithm developed in
Karlis (2002) that can also be used to perform this estimation. Once
[alpha], [beta] and [delta] have been estimated, the parameter [theta]
can be calibrated to verify the risk-neutral condition, that is,

H[[S.sub.T]; -[theta]] = [S.sub.0][e.sup.rT]. (23)

Alternatively, parameters [alpha], [beta], and [delta] can be
calibrated from market option prices through (22) and the parameter
[theta] can still be calibrated through the risk-neutral condition (23).
Both approaches yield similar results yet pricing using maximum
likelihood estimation is easier to implement.

It is worthwhile pointing out again that these parameters bring new
flexibility to the distortion (13). When calibrated, the resulting
distortion produces a risk-adjusted distribution that can be thought of
as including premiums for higher moments. In particular for skewness
through the parameter [beta].

GEOMETRIC BROWNIAN MOTION MODEL

The geometric Brownian model is used as a benchmark for Wang's
distortion. Prices are modeled by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [W.sub.t] is a standard Brownian motion. It is well known
that the price of a European call under this model is given by the
celebrated Black-Scholes formula. In Hamada and Sherris (2003), there is
evidence that the normal-based distortion of Wang (2000) can accurately
recuperate option prices under the Black-Scholes model. In order to
verify this, we simulate 1,000 log-normal prices [S.sub.T] with drift
[mu] = 16 percent, volatility [sigma] = 20 percent, and an initial price
of [S.sub.0] = 20. The maturity and the risk-free rate are respectively
set at T = 0.5 and r = 5 percent. The option prices are calculated for a
range of strike prices K going from 16 to 24.

In Table 1 the theoretical option prices are shown and compared
with options prices obtained through procedures (21) and (22). The
prices obtained with Wang's distortions (11) and (12) are
numerically equivalent to the Black-Scholes theoretical price as
expected. The prices obtained with the NIG distortion also reproduce the
numerical values obtained through the Black-Scholes formula. This first
test shows that all three distortions perform well under Gaussian
conditions. For the sake of completeness, in Table 2 we present results
for a put option. These results are computed using the call-put parity
relation. We remark that both distortions (NIG and normal) produce
similar results and, in fact, different simulation seeds produce results
where Wang's distortion can be slightly better than the NIG
distortion or viceversa. This is not unexpected since we know that
Wang's distortion replicates the Black-Scholes formula under a
geometric Brownian motion model.

JUMP DIFFUSION MODEL

The geometric Brownian motion produces continuous sample paths with
probability one. This feature can be considered too restrictive when it
comes to modeling certain types of assets, such as stocks (see Merton,
1976). In Merton (1976) we find a very simple model that incorporates
jumps in sample paths of the asset price. In Merton's model, the
price of the asset is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [mu], [sigma], [lambda], [mu]J, and [sigma]J are constants;
[W.sub.t] is a standard Brownian motion; [N.sub.t] is a Poisson Process
with parameter [lambda; and {[Y.sub.i]} is a sequence of Gaussian random
variables with mean [mu]J and standard deviation [sigma]J. Variables
[N.sub.t] and [Y.sub.i]s are assumed to be all independent. Notice that
this is the parametrization used in Hull (2009).

Merton's jump diffusion model produces an incomplete market
where the equivalent martingale measure is not unique. However, there
exists a martingale measure which only shifts the drift parameter [mu]
to r, and keeps the rest unchanged (in particular the distributions of
the jump times and sizes are identical under this martingale measure).
This mean-correcting martingale measure is used to calculate an
arbitrage-free price for a call option which is given by Merton (1976)
as

In Table 3 the theoretical option prices are shown and compared to
options prices obtained through procedures (21) and (22). It can be seen
that the prices obtained with the NIG distortion (13) are much closer to
the theoretical arbitrage-free prices than those obtained with
Wang's distortion (11). Indeed, the relative errors are smaller for
the prices given by the NIG distortion. We can also observe that it
performs as well as the Student-t distortion (12). This is a first
example that shows that Wang's distortion performs poorly in a
non-Gaussian context, whereas the Student-t and NIG operators are
flexible enough to price options accurately. For the sake of
completeness, in Table 4 we present results for a put option. These
results are computed using the call-put parity relation.

CEV MODEL

In the literature, there is evidence that assets log-returns are
heteroskedastic; that is, the variance of the log-returns is not
constant in time (Black, 1975). Because geometric Brownian motion
produces homoskedastic log-returns, an alternative model has been
proposed by Cox and Ross (1976) to capture this heteroskedasticy
feature. It is called the CEV model. In this model, the asset price is
given by the solution of the stochastic differential equation

where [W.sub.t] is a standard Brownian motion. We can recuperate
the Black-Scholes model from the CEV by posing [beta] = 2. If [beta]
< 2, volatility is a decreasing function of the asset price, which
causes a heavy left tail in the distribution of the asset. Conversely,
when [beta] > 2, volatility is an increasing function of the asset
price, which creates a heavy right taft. This feature can be useful to
capture the asymmetry in the log-returns distribution. Motivated by the
fact that there seems to exist an inverse relation between stock prices
and their volatility (Beckers, 1980) we chose to illustrate this model
with [beta] < 2. Under the CEV model, when [beta] < 2, the density
formula of the asset price is known in closed form (Schroeder, 1989).
The formula for the unique arbitrage-free price of the call option under
the CEV model when [beta] < 2 is also found in Schroeder (1989) and
is given by,

This is the parametrization used in Hull (2009) with [beta] =
2[alpha]. Here, [chi-square] 2(x; y, z) denotes the noncentral
chi-square cumulative distribution function evaluated at x where y is
the number of degrees of freedom and z is the parameter of
noncentrality. Schroder (1989) gives a method to compute the noncentral
chi-square cumulative distribution. Alternatively, the function dchisq
in R can also be used to readily compute this distribution.

We simulate 1,000 asset prices [S.sub.T] with [mu] = 15 percent,
[sigma] = 30 percent, [beta] = 1.5, and an initial price of [S.sub.0] =
20. The maturity and the risk-free rate are respectively set at T = 0.5
and r = 5 percent. The simulation was made using Milstein's method
(Higham, 2001), which is a simulation technique based on a
discretization scheme used to approximate Ito stochastic differential
equation solutions. The option prices have been calculated for a range
of strike prices K going from 16 to 24.

In Table 5 the theoretical option prices are shown and compared
with options prices obtained through procedures (21) and (22). It can be
seen that for in-the-money calls, the prices obtained with the NIG
distortion and Wang's distortion are close, and the relative errors
are small. We can also observe that the Student-t and the NIG operators
show an overall better performance than the normal distortion. Once
again, this is another situation where the NIG operator is more
efficient than Wang's distortion to do the call pricing. For the
sake of completeness, in Table 6 we present results for a put option.
These results are computed using the call-put parity relation.

EXPONENTIAL NIG-LEVY MODEL

The last model on which all three distortions are tested is the
exponential NIG-Levy model previously described in (4). The
arbitrage-free call price is given by (8). Once more, 1,000 asset prices
[S.sub.T] are simulated with [alpha] = 9, [beta] = 7.8, [delta] = 0.5,
[mu] = -0.7, and an initial price of [S.sub.0] = 20. The maturity and
the risk-free rate are respectively set at T = 0.5 and r = 5 percent.
The option prices are calculated for a range of strike prices K going
from 16 to 24.

In Table 7 the theoretical option prices are shown and compared to
options prices obtained through procedures (21) and (22). The results of
the simulation show that the NIG distortion over-performs Wang's
distortion for every strike price. The relative error for NIG distortion
prices remain smaller than 6.5 percent, while it can climb up from 13
percent to 34 percent for Wang's distortion. We can also observe
that the Student-t operator performs poorly as well when compared to the
NIG distortion in this case. For the sake of completeness, in Table 8 we
present results for a put option. These results are computed using the
call-put parity relation.

CONCLUSIONS AND FURTHER DISCUSSION

In this article we propose a generalized version of the distortions
proposed in Wang (2000, 2004). This generalization uses a NIG
distribution instead of a standard normal. All three distortions have
been tested using simulation. This empirical analysis using simulation
attempts to replicate non-Gaussian conditions as they could be found in
market data. It turns out that the NIG operator performs well for
various non-Gaussian models for the underlying asset whereas Wang's
operators can sometimes provide poor estimations of the theoretical
price.

The NIG distribution is a skewed distribution that has been proven
to effectively fit financial log-returns. It comes as no surprise that a
distortion operator based on this distribution performs better than
Wang's under non-Gaussian conditions. An interesting feature of
this distortion is that since it is based on a skewed distribution, it
distorts differently the right and the left tails of the underlying
distribution just like the Student-t distortion in Wang (2004). The
advantage is that this asymmetry can be controlled through the parameter
[beta] and a greater variety of shapes are possible. In this article we
set out to compare these three distortions in an option pricing context,
further testing is needed in order to compare the performance of these
distortions in different applications.

As for potential applications, this research contribution joins the
discussion in the literature regarding the connection between risk
measures and skewed fat-tailed distributions in finance. Indeed, there
have been several attempts to incorporate these stylized features of
asset returns into risk measures (e.g., Bali and Theodossiou, 2008). In
this article, we proposed a distortion-based risk measure that also
incorporates these features. The proposed NIG distortion produces a
distortion-based risk measure that has the same potential applications
as the normal-based risk measure in Wang (2000). In fact, there is a
wide range of possible applications that are now open to further
analysis and comparison. Concrete illustrations are needed in order to
test the NIG distortion and its performance with respect to other risk
measures. In this article, we limit ourselves to give a formal
construction as well as an empirical study of the NIG-based distortion.
We are yet to test its performance in those instances where the
normal-based and Student-t-based distortions have been put to use. This
will be the subject of future studies. Here, we only mention one of many
possible applications that are worth looking into with our newly defined
distortion.

In Wang (2004), the author carries out an empirical analysis
illustrating how yield spreads in catastrophe bonds can be modeled with
the distortions given by (11) and (12). Similar examples can be tested
with our NIG distortion. It would be interesting to compare the
performance of our proposed distortion in the same context instead of
using simulated data. In Wang (2004), the author uses yield spreads from
transaction data for catastrophe bonds. He then calibrates the
parameters [alpha] and k in the distortion (12) to these data set. The
resulting model seems to explain the observed yields in the
transactions. One possible direction for a future empirical study would
be to compare all three distortions (11), (12), and (13) with a similar
catastrophe bonds data set. The aim would be to calibrate all four
parameters to similar data sets and compare the performance of all three
models in describing the observed yield spreads. Such a study will
illustrate how risk-adjustment for higher moments is achieved through
all four parameters in the NIG distortion.

In general, our distortion (13) produces a family of risk measures
through (9) and as such it can be applied in different settings ranging
from capital allocation to optimal reinsurance. An interesting direction
to be explored in future research would then be to test the performance
of these newly defined family of NIG-based risk measures in those
applications where general distortion-based risk measures have been
used. In particular, a potential application of interest could be in the
problem of optimal reinsurance. In the literature, we recently find
studies that address this issue using risk measures (see, e.g., Bernard
and Tian, 2009). Other relevant references that give a review of
existing distortion risk measures and some of their applications in
insurance are Dowd and Blake (2006) and Balbas, Garrido, and Mayoral
(2009). Future empirical research can focus on applying the NIG-based
distortion risk measure in the context discussed in the above mentioned
articles.

Frederic Godin is with the University of Montreal. Silvia Mayoral
is with the University Carlos III of Madrid. The research of this author
was partially funded by the Ministerio de Ciencia e Innovacion
(ECO2009-10796). Manuel Morales is at the Department of Mathematics and
Statistics, University of Montreal. Manuel-Morales can be contacted by
morales@dms.umontreal.ca. This research was funded by the Natural
Sciences and Engineering Research Council of Canada (NSERC) operating
grant RGPIN-311660 and by Le Fonds quebecois de la recherche sur la
nature et les technologies (FQRNT) operating grant NC-113809